The Sphere Packing Problem in Dimension 4

We prove that the optimal sphere packing density in $\mathbb{R}^4$ is $\Delta_4=\pi^2/16$, achieved uniquely (up to isometry) by $\sqrt{2}\,D_4$, for all packings including non-periodic ones. The proof reduces to a Voronoi-cell lower bound $\mathrm{vol}(V)\ge 8$ via shell localisation, support-function containment for root-aligned configurations, and a coordinatewise-monotonicity reduction to a finite atlas of $176$ Weyl-orbit chambers.